The distribution function of the sum $Z$ of two standard normally distributed random variables $X$ and $Y$ is computed with the concept of copulas to model the dependency between $X$ and $Y$. By using implicit copulas such as the Gauss- or t-copula as well as Archimedean Copulas such as the Clayton-, Gumbel- or Frank-copula, a wide variety of different dependencies can be covered. For each of these copulas an analytical closed form expression for the corresponding joint probability density function $f_{X,Y}$ is derived. We apply a numerical approximation algorithm in Matlab to evaluate the resulting double integral for the cumulative distribution function $F_Z$. Our results demonstrate, that there are significant differencies amongst the various copulas concerning $F_Z$. This is particularly true for the higher quantiles (e.g. $0.95, 0.99$), where deviations of more than $10\%$ have been noticed.

翻译：通常分配的两种标准随机变量美元和美元美元之和的Z$和美元之差的分布功能,是用用于模拟X美元和Y美元之间依赖性的合金概念来计算。我们用一个数字近似算法来评价累积分配函数的双重组成部分,我们的结果表明,各种合金之间在美元方面有很大差异。对于较高方(例如0.95美元,0.99美元)来说,情况尤其如此,因为人们注意到这些高方位(例如0.95美元,0.99美元)的偏差超过10美元。